Calculator Inputs
Formula Used
The Euler beta function is
B(a,b) = ∫₀¹ t^(a−1) (1−t)^(b−1) dt = Γ(a)Γ(b) / Γ(a+b)
The incomplete and regularized forms are
Bₓ(a,b) = ∫₀ˣ t^(a−1) (1−t)^(b−1) dt
Iₓ(a,b) = Bₓ(a,b) / B(a,b)
This calculator uses a Lanczos approximation for Γ and a continued fraction for Iₓ(a,b), with an optional adaptive Simpson integral as a numerical check.
How to Use
- Pick Complete for B(a,b) via gamma relations.
- Pick Incomplete or Regularized and set x (0 to 1).
- Enter positive a and b for robust physics-style integrals.
- Enable the integral cross-check when exploring edge cases.
- After computing, export the result as CSV or PDF.
Example Data Table
| a | b | x | B(a,b) | Ix(a,b) | Bx(a,b) |
|---|---|---|---|---|---|
| 0.5 | 0.5 | 0.5 | 3.1415926536 | 0.5 | 1.5707963268 |
| 2 | 3 | 0.5 | 0.0833333333 | 0.6875 | 0.0572916667 |
| 2.5 | 3 | 0.6 | 0.0507936508 | 0.7529079625 | 0.0382429441 |
| 5 | 1.5 | 0.8 | 0.0738816739 | 0.5055606488 | 0.037351667 |
These examples are generated by the same engine used in the calculator, so you can compare your runs against known parameter sets.
Built for careful numeric work; validate extreme regimes.